# Differential Equations

## First Order Equations

Differential equations are widely used to describe a variety of dynamic processes in Economic Sciences, Business and Marketing. Below we consider how an advertising campaign can be simulated using differential equations.

Imagine that a company has developed a new product or service. The marketing strategy of the company involves aggressive advertising. To construct the simple model we introduce two variables:

• The quantity q (t) represents advertising activity that is described by spending rate, for example, by the amount of dollars (euros, pounds) the company spends for advertising per week;
• The quantity A (t) represents awareness of the target group of potential consumers about the new goods or services.

Thus, we will consider the market niche as a black box (Figure $$1$$). The advertising activity $$q\left( t \right)$$ is the input variable, and awareness of consumers $$A\left( t \right)$$ is the output variable that measures response of the system to the advertising campaign.

A simple model of such type was proposed in $$1962$$ and is called the advertising model of Nerlove and Arrow (the N-A model). This model relates advertising activity $$q\left( t \right)$$ and awareness of consumers $$A\left( t \right)$$ and is given by the differential equation:

$\frac{{dA}}{{dt}} = bq\left( t \right) - kA,$

where $$b$$ is a constant describing advertising effectiveness, $$k$$ is a constant corresponding to decay (or forgetting) rate.

The given equation contains two terms in the right side. The first term $$bq\left( t \right)$$ provides the linear growth of awareness of consumers as a result of advertising. The second term $$-kA$$ reflects the opposite process, i.e. forgetting about the product. In the first approximation, we can assume that the forgetting rate is proportional to the current level of awareness $$A.$$

This equation is a Linear Differential Equation of First Order. It's convenient to rewrite it in the standard form:

$\frac{{dA}}{{dt}} + kA = bq\left( t \right).$

The integrating factor is the exponential function:

$u\left( t \right) = {e^{\int {kdt} }} = {e^{kt}}.$

Therefore the general solution of the given differential equation is given by

$A\left( t \right) = \frac{{b\int {{e^{kt}}q\left( t \right)dt} + C}}{{{e^{kt}}}}.$

As usual, the constant of integration $$C$$ can be found from the initial condition $$A\left( {{t_0}} \right) = {A_0}.$$

In the examples below we consider how awareness of customers $$A\left( t \right)$$ varies for different advertising modes.

See solved problems on Page 2.